Lecture SS 2016

Lecturer: Josef Schicho, email: josef.schicho@risc.jku.at

Time and place: to be agreed.

Next meeting: 29.6.2016, 10:15, meeting room SP 416.

**Homework:** until 6.4..

**Homework:** until 20.4.. Please choose either (a) or (b).

**Homework:** until 28.4..

**Homework:** until 12.5..

**Homework:** until 25.5..

**Homework:** until 15.6..

**Homework:** until 29.6..

A linkage consists of several rigid parts, called links, whose relative position is determined by joints. For instance, the links of a planar linkage are polygons, and if two links are joined by a common vertex then the relative positions are related by a rotation around the common point. In rigidity theory, one is interested in the minimal number of links required so that some particular linkage does not move. Mathematically, a planar linkage is detemined by an undirected graph whose edges are labelled by positive real numbers. A realization is a function from the set of vertices to the plane such that the distance of two points connected by an edge is equal to the label of that edge.

The starting point of rigidity theory is the paper On graphs and rigidity of planar skeletal structures. It characterizes graphs such that a generic assignments of labels makes gives a rigid linkage (also known as Laman graphs), which means that the number of realizations up to translations and rotations is finite. We (i.e. the participants of the lecture and the lecturer) plan to address the following open questions:

Given a Laman graph, what is the number of essentially different realizations?

Given a Laman graph, can we find necessary/sufficient conditions on the labels for rigidity?

There are several approaches to these problems. Laman's inductive proof is elementary and explicit, and provides a concrete approach. Techniques from algebraic geometry/intersection theory are useful. Bond theory might be useful, too.

**Prerequisites:** As it is planned to discuss various approaches to the same type of problem, and some
approaches are elementary, knowledge in linear algebra and complex numbers are sufficient to understand the problem
and to pursue the more elementary approaches. A background in algebraic geometry or in
kinematics or in combinatorics is useful for other approaches,
but it is not required.